## Tuesday, March 11, 2014

### Range Building with Bluff Catchers

Figuring out how to apply poker theory to the practical strategic decisions that you run into day in day out when you're grinding at the tables can be quite difficult, so in this post I want to take a look at using GTORangeBuilder to analyze a specific type of situation that used to give me trouble when I was first learning Heads Up.

The keys to being able to actually apply game theory to improve your poker game are to identify a situation where you know strategy is weak. I find that the easiest way to spot this situations are scenarios where either:
1. I feel like my hand is face up and I am being taken advantage of or am being forced into complex "he knows that I know that my hand is face up" type of leveling mind games.
2. I’m folding not because I think you have the worst hand but because I can’t think of a reasonable line to continue with.
3. I regret my previous action because I feel like it just gave my opponent a chance to exploit me later in the hand.

Once you've identified a leak in your strategy, you just need to think about what your range is and what your opponents range is at various points in the hand and look for clear weaknesses.  Then you can start to figure out out how to adjust your play to plug the leak.

GTORangeBuilder is designed to take care of determining how to optimally plug the leak for you, while identifying the situations where you have leaks is something you’ll have to do yourself as you review past sessions.

The call-check-donk line.

I’m going to run through an example that gave me problems when I was just learning to play HU. These days the solution I’m describing is one of several standard plays that can combat the call-check-donk line, but GTORangeBuilder lets us take a look at the specific mathematical benefits of this “standard play” and my hope is that this example will help people identify and solve new types of problems that can come up when you have unbalanced ranges.

I first started learning to play HU poker in 2008-2009 and to my surprise I found myself struggling in certain spots against players that seemed quite fishy, especially given the way the game was played at the time.

The line that gave me trouble was what I call the call-check-donk line. I’d be playing a generally loose passive opponent who would almost always defend his big blind, say with 90% of hands. I’d c-bet aggressively and he’d almost never fold (c-betting very aggressively here isn't necessarily the best choice, but it's what I tried). If I checked the turn, he’d then lead out with a very wide range on the river.

I viewed this player as a fishy calling station and was trying to play exploitatively against him by value betting very thinly and by betting big so as to get paid off whenever I had a hand. This meant that I was betting big on the turn with almost any strong or medium strength hand to “exploit” his passive tendencies.

However, by doing this my range for checking back the turn was so unbalanced that even a generally fishy player was able to completely exploit me on the river. By betting the turn with all of my medium and strong hands, any time I got to the river without betting the turn my hand was face up as weak and my opponent would donk-bet the river at me and take down the pot.

This made me feel like I couldn't check the turn with air without completely giving up on the hand. Furthermore, my opponents river donking range was actually extremely well balanced, as they would be check/calling the flop with lots of stronger hands as well as lots of air, and draws.

I was so focused on exploiting my opponent that I didn't pay attention to the massively exploitable leak that I had introduced into my own game. At the time, the way people thought about poker, particularly against fish, didn't involve much consideration of what your own range was in specific situations.  People tended to focus entirely on exploiting their opponents weaknesses with specific hands without realizing that even fishy players can punish you quite effectively if you unbalance your ranges too drastically in an attempt to exploit them maximally.

The easiest way to solve this river problem is to check back some of your medium strength hands on the turn so that your river range is at least balanced between some air and some medium strength hands to use as bluff catchers. This generally lets you get just as much value out of your bluff catchers as you would by barreling the turn with them and also lets you see cheap show downs with the stronger portions of your air. The synergy between your bluff catchers and your strong air on the river is a perfect example of how building strong ranges can drastically increase your EV in everyday situations.

With GTORangeBuilder we can actually take a look at the effects of adding a bluff catcher to your river range on your expected values both with your air and with your bluff catchers. This lets us quantitatively measure the EV of leveraging the synergies between your stronger air that wants a free showdown (eg A high) and your bluff catchers that are ahead of your opponents floating range (eg middle pair).

In this example I’m going to look at two river scenarios. In the first scenario, I’ll consider the hero, who is in position, getting to the river with a range that is pure air vs a villain with a balanced range of air, missed draws, and strong hands. In the second scenario I’ll take the exact same villain range, and I’ll add a medium strength bluff catcher to the hero’s range. The key thing to observe is that by adding a bluff catcher to the hero range, not only does the hero get a strong EV from that bluff catcher in isolation, but it also significantly increases the hero’s ev with all the other hands in his range.

To keep the examples simple and easy to understand, rather than putting in complete ranges for each player I put in simplified “representative” ranges. A representative range is a range that has the right types of hands in approximately the right proportions, rather than actually listing every single hand in a players ranges. As an example, rather than considering all 192 combos of Ax as separate hands we might break that range into two representative hands of Ad6s and AdQs on a non-flush board so that we can focus strategically on how to play a strong ace vs a weak ace. For a human player, trying to play a strategy that varies its play with all 192 Ax combos isn’t really feasible so using representative ranges is a good way to highlight the strategic differences between hands in a way that you can learn from and apply to your games.

In both the scenarios below, we assume the action went as follows:
1. Hero raises preflop to 3x, villain calls.
2. Villian check calls a bet of 4bb on the flop.
3. Both players check the turn.
To keep things simple, I also assume that no one is going to do anything too crazy on the river (such as putting in > 40 bbs running a huge bluff) and again for simplicity, I considered a small range of bet sizes.

Scenario 1: Hero has only air. Villian has busted draws, random floats, and made hands.

Pot: 14bb
Board: Kc8d2d2h4s
Hero Range: 7c5c, Ah9h, QsJh
Villian Range: 6h5h, Td9d, Qc7h, 8s7d, Ks4c

Scenario 2: Hero adds a medium strength bluff catcher to his range

Pot: 14bb
Board: Kc8d2d2h4s
Hero Range: 7c5c, Ah9h, QsJh, Qh8h
Villian Range: 6h5h, Td9d, Qc7h, 8s7d, Ks4c

The first thing worth noting about these ranges is that the hero actually has almost 50% equity in scenario 1, simply due to both players having air quite often. However, because the hero’s range is poorly constructed he is unable to actually defend his equity and win his fair share of the pot, despite being in position. In fact, as you can see in the solution browser below, even with the top of the hero’s range (Ah9h) in scenario 1 the hero only gets about 40% of the pot on average (click on the left most node of the tree and look at Ah9h in the "Hero Range" chart to see this). Over all the Hero's EV in this scenario is 3.33bb out of the total 14bb pot when both players play optimally.

In scenario two, Qh8h has about 70% equity vs the villains range and manages to average winning close to 70% of the 14 chip pot with that hand.  However, what is more powerful is that adding Qh8h to the players range increases his EV with all of his other hands.  This means the total EV that we gain by putting Qh8h into our river range is the 9.16bb we earn when we have Qh8h plus the additional EV that we gain by increasing our expectation with our other hands.  Even if we ignore the cases when we actually have Qh8h, our average EV with our other hands has increased from 3.33bb in scenario 1 to 4.19bb.  On average we will win almost an extra bb every single hand with the weaker parts of our range just by making our entire range more defensible!  In a game like HU where this type of river situation can easily happen once out of every 10 hands, this type of adjustment could have a 10bb/100 impact on our winrate.  We've also managed to take a line that does at least as well with Qh8h as betting the turn is likely to.

You can browse scenario 2 in the solution browser below.  As you can see the villain is forced to play significantly more passively when 25% of the hero range is a medium strength bluff catcher.

Furthermore, the hero can now play more aggressively.  Because the villain has to check back a lot of his air in scenario 2, the hero can turn 7c5c into a bluff and take a bet-check-bet line more often (almost twice as often, 53.3% vs 30% according to the solutions). Shout out to NoahVanderoff from reddit for pointing this out.

This example is designed the highlight the key concept at the heart of building optimal ranges.  Ranges that lack the proper diversity can make it extremely difficult for us to defend our equity against a competent opponent, even when we play optimally.  We cannot just try and "out think" or "out play" an opponent when we have a poorly constructed range unless we can rely on him making a lot of major mistakes.  The much better response is to instead adjust our hand range by adding synergistic hands that perform well on there own while also improving the performance of all other hands in our range.

Submit your own scenarios for analysis

If you have a leak or river scenario that you'd like help analyzing post it in the comments and if I have time, I'll analyze it in depth and make it the topic of a future blog post.